A rooted tree is an ordinary tree with an equivalence condition: two trees
are the same if and only if one can be transformed into the other by reordering
subtrees. In this paper, we construct a bijection and use it to generate rooted trees
(or forests) of any specified nodecount m uniformly at random. As an application,
Raddum and Semaev  Raddum and Semaev propose a technique to solve systems
of polynomial equations over 2 as occurring in algebraic attacks on block ciphers. This approach is known as MRHS. In  Geiselmann, Matheis, and Steinwandt propose an ASIC hardware design to implement MRHS, and they show that the use of ASICs seems to enable significant performance gains over a software implementation of MRHS. What hasn't been asserted is the total time complexity of their platform, though individual components' runtimes are provided. If one supposes that deletions in MRHS occur as rooted trees generated uniformly at random, then one application of the proposed algorithm would be to contribute to such a time complexity; experiments are generated to provide statistical averages of key quantities.
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